1. Field of the Invention
The present invention relates to a method for computing the covariance between time series of losses, profitability, or other appropriate metrics on various product segments within a retail portfolio for use in portfolio optimization and economic capital calculation.
2. Description of the Prior Art
Portfolio optimization and economic capital calculation are key strategic issues for retail portfolio managers. Portfolio optimization refers to creating an optimal blend of products and services across different consumer segments. Computing economic capital seeks to determine how much capital a bank needs to hold in order to protect against an economic downturn in which a large number of loans default in the same time period. If the calculations for either the portfolio optimization or the economic capital calculation are not correct, the bank risks: booking the wrong proportion of loans; not correctly pricing for the risks assumed; or being inefficient in their allocation of capital.
Portfolio optimization and economic capital calculation are known to be treated together because the considerations for both are similar. More particularly, for portfolio optimization, the expected profit for each product segment; the profit volatility within each segment; and the correlations between segments, must be accurately predicted. For economic capital calculation, the expected losses for each product segment; the loss volatility within each segment; and the correlation of losses between segments must be accurately predicted.
Harry Markowitz initiated modern portfolio theory with his landmark paper in 1952; “Portfolio Selection”, H. M. Markowitz, Journal of Finance, Vol. 7, Issue 1, pages 77-91, 1952, hereby incorporated by reference. The question presented in the paper was essentially, “How is the optimal investment blend chosen across a set of different opportunities?” Mr. Markowitz theorized that the optimal portfolio is one which maximizes return while simultaneously minimizing the volatility of returns.
The ratio of return over volatility is referred to as the Sharpe Ratio. For a set of investment opportunities,
      S    i    =                    r        i            -              r        f                    σ      i      where Si is the Sharpe Ratio for the ith investment opportunity, ri is the expected return, σi is the expected volatility, and rf is the risk-free rate of return, such as from a US government bond. In retail lending, return is the expected profitability for the loan or pool of loans and the volatility is the uncertainty in obtaining the anticipated margin. The portfolio's hurdle rate should be substituted for the risk-free rate. Hurdle rates are described in detail in: Principles of Corporate Finance, Fifth Edition”, by Richard A. Brealey and Stewart C. Myers, McGraw-Hill, 1996, hereby incorporated by reference.
Markowitz also recognized the importance of the correlation between investment opportunities in his solution. For example, in retail lending, Markowitz's theory would say that it would not be sufficient to determine that five (5) mid-tier credit card segments all had the highest profit-volatility ratio because those segments are likely to move in unison relative to the economy. According to Markowitz's theory, placing the entire portfolio's growth in those five segments would thus not be optimal because it does not confer any diversification benefit.
The idea behind a diversification benefit is that the less correlation there is between investments, the lower the overall volatility of the portfolio. For example, considering two products that provide equal returns but are highly anti-correlated, the best answer is to invest in both equally. The average return would be unchanged, but the net volatility would be much lower.
Markowitz provided a simple solution to this optimization problem. Subsequent work incorporated the notion of business constraints so that more realistic solutions could be derived. Many commercially available software packages are available to compute the ideal portfolio blend using Markowitz's theory, as long as the user provides the expected profit and volatility for each instrument, and covariance between instruments. Examples of such commercially available software packages are “S+” and “SAS”.
Unfortunately, many portfolio optimization implementations, such as those mentioned above, use historical averages for the rate of return. Taking such a backward-looking view of the portfolio has led to some of the major portfolio failures of the last decade. If no reliable forecasts are available, portfolio optimization should not be attempted and capital calculations will be badly skewed. Because of the difficulties in computing expected returns and expected volatility, modern portfolio theory, as promulgated by Markowitz, has often proven unreliable and even dangerous when applied to market-traded instruments.
However, retail portfolios are predictable if the unique dynamics of those portfolios are taken into consideration, as discussed in co-pending commonly owned U.S. patent application Ser. No. 09/781,310, filed on Feb. 13, 2001, entitled “Vintage Maturation Analytics for Predicting Cash Flow for Customer Commodities and their Responses to Economic, Competitive, or Management Changes” and U.S. patent application Ser. No. 10/359,895, filed on Feb. 7, 2003 and published as US Patent Application Publication No. US 2003/0225659 A1 on Dec. 4, 2003, entitled “Retail Lending Risk Related Scenario Generation”, hereby incorporated by reference. The technology disclosed therein to create accurate forecasts of profit (or loss) and profit volatility (or loss volatility) is called Dual-time Dynamics (DtD).
Briefly, DtD decomposes historical data into tenure-based, time-based, and vintage-based components. FIG. 1 shows a top down view of retail portfolio data. Each month a new vintage is booked. A vintage is a group of accounts all booked in the same period of time. Each of the diagonal lines in FIG. 1 represents the aging of a vintage in both calendar date and months-on-books (tenure). The performance of the vintage in profitability, loss rate, or any other metric, would be represented by a line coming out of the page. DtD analyzes all of the vintages simultaneously according to the following model: r(ν,a,t)=H(ƒm(a),ƒg(t),ƒq(ν)), where r is the rate being modeled, v is the vintage, a is the age (months-on-books) of the vintage, t is the calendar date, ƒm(a) is the maturation curve measured either parametrically or non-parametrically as a function of months-on-books, ƒg(t) is the exogenous curve measured non-parametrically as a function of calendar date, ƒq(ν) is the vintage quality curve measured as a non-parametric function of vintage, and H is the composition function of the curves to create the final answer. If H is additive, then r(ν,a,t)=ƒm(a)+ƒg(t)+ƒq(ν), or when H is multiplicative then r(ν,a,t)=ƒm(a)·ƒg(t)·ƒq(ν).
Nonlinear functions are also frequently used for H. DtD takes the vintage performance data as in FIG. 1, assumes a form for H, and then estimates the functions ƒm(a), ƒg(t), and ƒq(ν) by solving the inverse problem. The method of generalized additive models, for example, as described in “Generalized Additive Models”; by T. J. Hastie and R. J. Tibshirani, published by Chapman and Hall, New York, 1990, uses a similar method of solution, except with purely orthogonal functions and an additive H. The maturation curve captures the product lifecycle cleaned of noise from the economy or marketing plans. The exogenous curve measures primarily the seasonality and macroeconomic impacts. Vintage quality measures the intrinsic risk of the vintage cleaned of variations in the economy. Credit scores are reasonable predictors of vintage quality.
The exogenous curve is further analyzed using standard econometric methods to extract seasonality and macroeconomic impacts (called the trend). The full decomposition process is shown in FIG. 2 and described in detail in the patent applications mentioned above.
Once the three components, i.e. tenure-based, time-based and vintage-based components, have been derived, forecasts are created by inserting a scenario for the future of the macroeconomic impacts on the exogenous curve. This is combined with the seasonality, vintage quality, and maturation curve to create a forecast for each vintage. For future vintages, a scenario for their quality must be included. The shaded region of FIG. 1 represents the area to be forecasted and the scenario elements required to create the rate forecasts. FIG. 3 illustrates how those forecasts are combined with scenarios for the volume of new bookings and other rate forecasts to create the final portfolio forecast. When modeling in the context of economic capital, the key rates are usually PD (probability of default), EAD (exposure at default), and LGD (loss given default), as described in detail in the Basel II literature, for example, “International Convergence of Capital Measurement and Capital Standards”, by the Basel Committee on Banking Supervision, Bank for International Settlements Press & Communications, CH-4002, Switzerland, Copyright 2005, pages 1-284. To model profitability, additional variables for revenue generation and attrition are usually included, for example, as defined in detail in Principles of Corporate Finance, Fifth Edition”, by Richard A. Brealey and Stewart C. Myers, McGraw-Hill, 1996, hereby incorporated by reference.
Expected Volatility
Co-pending, commonly owned U.S. patent application Ser. No. 10/359,895, filed on Feb. 7, 2003 and published as US Patent Application Publication No. US 2003/0225659 A1 on Dec. 4, 2003, entitled “Retail Lending Risk Related Scenario Generation”; shows that the outputs of DtD can be combined with a Monte Carlo scenario generator to compute the expected volatility for the portfolio. FIG. 4 shows the process whereby the previously measured exogenous trend is analyzed to create a predictive ARIMA-style model, for example, as described in “Applied Econometric Time Series, 2nd Ed.”, by Walter Enders, Wiley Series in Probability and Statistics, 2004. hereby incorporated by reference and a distribution of model residuals. From this, scenarios for ƒg(t) are created as: {tilde over (ƒ)}g(t)=y(ƒg(t−1))+εt, where y is an econometric model like ARMA or ARIMA, ƒg(t−1) is the function of the exogenous trend for all times up to t−1, and εt is the noise term taken from the econometric model residuals. These scenarios are fed through the forecasting procedure shown in FIG. 3 to produce a distribution of future profit or loss. The distribution can be measured at one-standard deviation for profit volatility or at some higher level (99.9% in the Basel II specification) to measure economic capital.
Co-pending commonly owned U.S. patent application Ser. No. 09/781,310, filed on Feb. 13, 2001, entitled “Vintage Maturation Analytics for Predicting Cash Flow for Customer Commodities and their Responses to Economic, Competitive, or Management Changes” and U.S. patent application Ser. No. 10/359,895, filed on Feb. 7, 2003 and published as US Patent Application Publication No. US 2003/0225659 A1 on Dec. 4, 2003, entitled “Retail Lending Risk Related Scenario Generation”, hereby incorporated by reference, demonstrate that the processes represented in FIGS. 1-4 are sufficient and accurate for forecasting and computing expected volatility. However, for portfolio optimization and diversified economic capital, the covariance must be measured. Covariance relates to the expected correlation between products. Measuring covariance is not a simple process, because the performance data is contaminated with marketing plans and product lifecycle impacts which cannot be expected to recur in the future.
For example, consider a scenario in which new products are launched in two different areas in the same quarter, say a prime auto product and a sub-prime mortgage product. Because of lifecycle effects present in both products, overall loss rates will rise steadily for the first several years regardless of what happens in the macroeconomic environment. Computing the correlation coefficient between the performance time series of two products would lead to an interpretation that they are very highly correlated. However, from this kind of management-induced correlation, is it reasonable to expect that prime auto will be highly correlated to sub-prime mortgage in the future?
There is a further complication of deciding which time series to correlate, since many different variables have an impact on the total portfolio performance. Thus, there is a need for accurately predicting the correlation between different product segments in order for use in portfolio optimization and economic capital calculation.